-modules in non-describing characteristic, part I
Fix a finite field . Let be the category of finite dimensional -vector spaces with injections, and let be a noetherian ring. We study the category of functors from to -modules in the case when the characteristic of is invertible in . Our results include a structure theorem, finiteness of regularity, and a description of the hilbert series.
Fix a finite field of cardinality , and a noetherian ring . Let be the th general linear group over . Roughly speaking, the aim of this paper is to study the behavior of sequences, whose th member is a -module, as approaches infinity (the “generic case”). As varies, every prime appears as a divisor of the size of . But surprisingly, it is possible to avoid most of the complications of the modular representation theory in the generic case after inverting just one prime, namely the characteristic of . So in this introduction, unless mentioned otherwise, we assume that is invertible in (we are in “non-describing” characteristic).
We obtain these sequences in the form of -modules. A -module is a functor , where is the category of finite dimensional -vector space with injective linear maps. Clearly, acts on . Thus can be thought of as a sequence whose th member is a -module. This sequence could be arbitrary if we do not impose any finiteness condition on . But there is a natural notion of “finite generation” in the category of -modules. This paper analyzes finitely generated -modules. Here is a sample theorem that we prove (it extends [GW, Theorem 1.7] away from characteristic zero, and also improves some cases of [SS5, Corollary 8.3.4]):
[-polynomiality of dimension] Assume that is a field. Let be a finitely generated -module. Then there exists a polynomial such that for large enough .
The result above is a consequence of our main structural result that we prove about finitely generated -modules. Given a -module and a vector space , we can define a new -module by
We call this new -module the shift of by . Our main result roughly says that the shift of a finitely generated module by a vector space of large enough dimension has a very simple description. To make it precise, note that there is a natural restriction functor
This functor admits a left adjoint . We call a -module induced if it is of the form for some . A -module that admits a finite filtration whose graded pieces are induced is called semi-induced. We now state our main theorem.
[The shift theorem] Assume that we are in the non-describing characteristic, and that is noetherian. Let be a finitely generated -module. Then is semi-induced if the dimension of is large enough.
1.1. Idea behind the shift theorem
The shift theorem is proven by induction on the degree of generation. To make the induction hypothesis work, we construct a “categorical derivation” in the monoidal category of Joyal and Street [JS]. To make it precise, let be the category of finite dimensional -vector spaces with bijective linear maps. Joyal and Street considered a monoidal structure111It is shown in [JS] that this category is actually a braided monoidal category if is a field of characteristic zero. But we don’t need the braiding, and so we don’t the characteristic zero assumption on given by
We construct a categorical derivation on . In other words, satisfies
As pointed out to us by Steven Sam, there is an algebra object in such that the category of -module is equivalent to the category of -modules. Under this equivalence, induced modules are -modules of the form . Our categorical derivation shows that if we apply the cokernel of to an induced module then we obtain another induced module of strictly smaller degree of generation. This is what makes our inductive proof work. But there is a caveat. Everything said and done in this paragraph so far is true without any restrictions on the characteristic. On the other hand, the shift theorem is false if we drop the non-describing characteristic assumption.
The category naturally contains a localizing subcategory whose members are called torsion -modules. Given a -module , we denote the maximal torsion submodule of by . The functor is left exact, and its right derived functor is denoted . A crucial technical ingredient in our proof of the shift theorem is the following criterion for semi-induced modules.
Let be a finitely generated -module. Then is semi-induced if and only if . That a semi-induced satisfies is easy to prove, and doesn’t require any assumptions on the characteristic. But the converse requires the non-describing characteristic assumption in two crucial and separate places: (1) is exact, and (2) commutes with . The first one is immediate from our construction of , but the second one requires an interesting combinatorial identity (which appears in the proof of Lemma 4.3.1).
We actually only use the following immediate corollary of this theorem, which provides us control over the torsion part of a module. In fact, our argument shows that, in the non-describing characteristic, the theorem above is equivalent to its corollary below.
Let be a finitely generated -module. Then if the dimension of is large enough.
All these ingredients above allow us to show by induction on the degree of generation that is semi-induced if is large enough. The shift theorem then follows from it.
1.2. Some consequences of the shift theorem
To start with, Theorem 1 is a consequence of the shift theorem simply because induced modules can be easily seen to satisfy -polynomiality of dimension. If we drop the non-describing characteristic assumption, and assume that , then defines a finitely generated -module. This implies that -polynomiality fails in equal characteristic, and so the shift theorem must also fail. Below we list some more consequences.
[Finiteness of local cohomology] Let be a finitely generated -module. Then we have the following:
For each , the module is finitely generated. In particular, if the dimension of is large enough.
for large enough.
The theorem above extends Corollary 1.1 to the higher derived functors of . We use this theorem, and an argument similar to the one for -modules as in [NSS1], to bound the regularity. In particular, we provide a bound on the regularity in terms of the degrees of the local cohomology.
[Finiteness of regularity] Let be a finitely generated -module. Then has finite Castelnuovo-Mumford regularity.
Gan and Watterlond have shown in [GW] that, when is an algebraically closed field of characteristic zero, then any finitely generated -module exhibits “representation stability” (a phenomenon described by Church and Farb [CF]). We recover their result in a much more systematic way (more precisely, we believe that one can also write down a virtual specht stability statement away from characteristic zero as done for -modules by Harman in [Har]). Below, we only state a part of the result to avoid giving a full definition of representation stability here.
[[GW, Theorem 1.6]] Assume that is an algebraically closed field of characteristic zero. Let be a finitely generated -module. Then the length of the -module stabilizes in .
We also obtain the following new theorem in characteristic zero.
[Finiteness of Injective dimension] Assume that is a field of characteristic zero. Then the following holds in :
Every projective is injective.
Every torsion-free injective is projective.
Every finitely generated module has finite injective dimension.
Along the way, we classify all indecomposable injectives in characteristic zero, and we also classify indecomposable torsion injectives when is an arbitrary noetherian ring.
1.3. Relations to other works
Recently, Kuhn in [Kuh] has analyzed a similar but simpler (of lower krull dimension) category of -modules, where is the category of finite dimensional vector spaces.
[[Kuh, Theorem 1.1]] In non-describing characteristic, is equivalent to the product category . In particular, if is a field then is of krull dimension zero.
It is a folklore that one recovers the representation theory of the symmetric groups from the representation theory of the finite general linear group over by setting . We observe a similar phenomenon between -modules and -modules: all the results we have for -modules in the non-describing characteristic are true for -modules in all characteristic (-modules encode sequences of representations of the symmetric groups; see [CEF]). In other words, the proofs for the results on -modules are degenerate cases of the proofs for the corresponding results on -modules in the non-describing characteristic. But we point out that (1) many of our ideas are copied from the corresponding ideas on -modules, and (2) we know a lot more about -modules, for example, all the questions that we pose below have been solved for -modules. We have tried to summarize throughout the text where each crucial idea has been borrowed from, but here is a list of references that contain analogs of our results – [Ch, CE, CEF, CEFN, Dja, DV, Li, LR, Nag1, NSS1, Ram, SS1].
1.4. Further comments and questions
Theorem 1.2 implies that every finitely generated object in the category
of generic -modules is of finite length, that is, the krull dimension of is zero. In a subsequent paper [Nag2], we shall prove that the same holds in the non-describing characteristic (where is still assumed to be a field) by providing a complete set of irreducibles of the generic category. Determining Krull dimension in equal characteristic () is related to an old open problem called the strong artinian conjecture [Pow, §2].
Sam and Snowden have proven that the categories of torsion and the generic -modules are equivalent in characteristic zero [SS1, Theorem 3.2.1], and such a phenomenon seem to appear in some other categories as well (for example, see [SS4] and [NSS2] for the category of -modules). We have the following question along the same lines:
Assume that is of characteristic zero. Is there an equivalence of categories .
Our result provides bounds on the Castelnuovo-Mumford regularity in terms of the local cohomology. But we have not been able to bound local cohomology in terms of the degrees of generation and relation. An analogous question for -modules has already been answered ([CE, Theorem A]); also see [Ch], [Li], and [LR, Theorem E] for more results on this. We also note that, in characteristic zero, Miller and Wilson have provided bounds on the higher syzygies for a similar category called -modules; see [MW, Theorem 2.26].
Let be a -module generated in degrees and whose syzygies are generated in degrees . Is there a number depending only on and such that for every vector space of dimension larger than .
Is it true that the Castelnuovo-Mumford regularity of a -module is exactly where varies over the finitely many values for which is nonzero.
1.5. Outline of the paper
In §2, we provide an overview of -modules. In particular, we sketch an equivalence between and the module category of an algebra object in the monoidal category of Joyal and Street, and we recall some formalism of local cohomology and saturation from [SS2]. In §3, we prove some formal properties of induced and semi-induced modules that we need. These properties are formal in the sense that they have nothing much to do with -modules and are true (with appropriate definitions) in several other categories (for example, , or ). We decided to include a short section and collect these formal results at one place. The meat of the paper is contained in §4 where we prove the shift theorem. The last section (§5) contains all the consequences of the shift theorem.
2. Overview of -modules
We work over a unital (not necessarily commutative) ring . For a non-negatively graded -module , we define to be the least integer such that for , and if no such exists.
We fix a finite field of cardinality , and assume that all vector spaces are over . For a vector space , we denote the group of automorphisms of by or . When the dimension of is , we also denote these groups by . We denote the trivial vector space by , and we shall simply write whenever .
2.1. The monoidal category of Joyal and Street
We denote, by , the category of finite dimensional vector spaces with isomorphisms. A -module is a functor from to . -modules form a category which is naturally equivalent to the product category . In particular, a -module is naturally a non-negatively graded -module. We denote, by , the -module satisfying
If , we say that is supported in degree . Given -modules we define an external product by
Then turns into a monoidal category; see [JS, §2].
2.2. The algebra
Let be the -module such that is the trivial representation of for each . We have a map given by
This turns into an algebra object in the monoidal category . We denote the category of -modules by . The -module is naturally an -module. As usual, the degree of generation of an -module is defined to be . We shall denote by , and so the degree of generation of is . We say that an -module is presented in finite degrees if and are finite.
2.3. Definition of a -module
We denote, by , the category of finite dimensional vector spaces with injective linear maps. A -module is a functor from to . We denote the category of -modules by . Let be a -module. A morphism induces a map which we denote by . The -module restricts to a -module and admits a natural map given by
where is the inclusion. Conversely, if is an -module and is a -morphism, then we have a map given by the composite
where the first map comes from -module structure on and the last map comes from -module structure on . It is easy to see that the above discussion describes an equivalence of categories:
is equivalent to .
We shall not distinguish between -modules and -modules. In particular, notions like degree of generation makes sense for -modules. We explain degree of generation from the perspective now. Given a -module , we can upgrade it to a -module by declaring that all -morphisms that are not isomorphisms acts on by . This defines a functor . We define to be the left adjoint to . Let be a -module. Denote the smallest -submodule containing for by . Then is given explicitly by
The functor (called -homology) is same as the functor under the equivalence above. We shall use the notation instead of . Here are some basic results on -homology.
We have . In particular, if then the natural map is just the inclusion map .
Let be a -module, and be a morphism of -modules. Then we have the following
if and only if .
is an epimorphism if and only if is an epimorphism.
Suppose and for . Then if and only if .
Part (a) is just the Nakayama lemma, and (b) follows from (a) and the right exactness of . For part (c), suppose . By part (a), it suffices to show that . First suppose is a vector space of dimension at most . Since for all , the map factors through the projection and is naturally isomorphic to . This shows that
Next suppose is a vector space of dimension bigger than . Since is a surjection and is right exact we see that . Thus , completing the proof. ∎
2.4. Local cohomology and saturation
Let be a -module. We say that an element is torsion if there exists an injective linear map such that . A -module is torsion if it consists entirely of torsion elements. We denote the maximal torsion submodule of by , the th right derived functor of by , and the degree of by . Let be the category of torsion -modules. It is easy to see that is a localizing subcategory. Let be the corresponding localization functor and be its right adjoint (the section functor). We define saturation of to be the composition . We denote the th right derived functor of by .
We refer the readers to [SS2, §4] where the formalism of local cohomology and saturation is discussed in quite generality. The authors needed an assumption to work out their theory which in our case is the following:
Injective objects of remain injective in .
The arguments in [SS2, §4] are valid if we replace (*) by the following alternate assumption:
If is injective then so is .
The injective hull of a torsion module is torsion. If is injective then so is . In particular, (**) holds.
Recall that injective hulls are essential extensions. Since an essential extension of a torsion module is torsion, the first assertion follows. Now since is injective and contains , contains the injective hull of . By the first assertion and the maximality of , we conclude that is injective. ∎
Let be an object of the right derived category which can be represented by a complex of torsion -modules. Then , and .
We now state a result from [SS2] that we need.
[[SS2, Proposition 4.6]] Let . Then we have an exact triangle
where the first two maps are the canonical ones.
We call a -module derived saturated if , or equivalently (see the proposition above), is an isomorphism in .
3. Induced and semi-induced -modules
The aim of this section is to prove some formal properties of induced and semi-induced modules. The restriction map admits a left adjoint denoted , which is exact. By definition of , we have the adjunction
We call -modules of the form induced. If is supported in degree we say that is induced from degree . Moreover, when is a -module isomorphic to then we denote by simply . By Yoneda lemma, we have . We have the following alternative descriptions for :
The composite functor is naturally isomorphic to the identity functor on -modules. The counit is an epimorphism on any -module.
The first assertion is clear because composing with yields , which is naturally isomorphic to the identity functor. Alternatively, by adjointness of and , we have
and so the result follows by the uniqueness of left adjoints. For the second assertion, it suffices to check that is faithful, which is trivial. ∎
A useful thing to note is that if is a -module and is a map of -modules then the image of the corresponding map is the smallest -submodule of containing the image of . In particular, if is surjective then is surjective.
is a projective -module if and only if is a projective -module. All projective -modules are of the form .
Each of and is left adjoint to an exact functor ( and respectively), so both of them preserve projectives ([Wei, Proposition 2.3.10]). Since (Proposition 3), we conclude that is projective if and only if is projective.
For the second assertion, let be a projective -module. By Proposition 3, there is a natural surjection , and since is projective it admits a section . Let be the map given by . It suffices to show that is an isomorphism. By Proposition 3, we have
Thus, by Proposition 2.3, is surjective. Since is projective we have a short exact sequence
In particular, . Thus, by Proposition 2.3, we conclude that is an isomorphism. This completes the proof. ∎
has enough projectives.
for and is isomorphic to for . In particular, , and is presented in finite degrees if and only if .
Let be -modules induced from . Then induces an isomorphism
whose inverse is given by .
Kernel and cokernel of a map of -modules induced from are induced from . An extension of -modules induced from is induced from .
Let be a map of -modules. Then by the previous proposition, there is a such that . Since is exact, we have and , proving the first assertion. For the second assertion, let be an extension of and . Let and be projective resolutions of and such that and are all supported in degree . By the horseshoe lemma and Proposition 3, is a projective resolution of . By the first assertion, is induced from . ∎
Let be a module induced from . And let be a submodule of generated in degrees . Then is isomorphic to . In particular, is induced from .
Since is generated in degree and for , we have . It follows that the natural map is a surjection (Proposition 2.3). Composing it with the inclusion , we obtain a map . By construction, is the natural inclusion . Thus by the Proposition 3, we have
This implies that is injective, completing the proof. ∎
Let be a -module. Then
is generated in degrees if and only if it admits a surjection with .
is presented in finite degrees if and only if there is an exact sequence
such that .
Proof of (a). Suppose there is a surjection . Since is right exact, we have a surjection . This shows that . Conversely, suppose . Let be the -module with satisfying for . By construction, we have a surjection . By Nakayama lemma, the natural map is a surjection, completing the proof.
Proof of (b). First suppose is presented in finite degrees. Then by part (a), there is a surjection with . It suffices to show that the kernel of is generated in finite degrees. But this follows from the long exact sequence corresponding to . Conversely, if there is an exact sequence
such that . Then by part (a), and the kernel of are generated in finite degrees. Again, the long exact sequence corresponding to finishes the proof (see Proposition 3). ∎
3.1. Semi-induced modules
We call a module semi-induced if it admits a finite filtration whose graded pieces (successive quotients) are induced modules that are generated in finite degrees.
Suppose and assume that is concentrated in degree . Then is induced from . In particular, is homology acyclic.
By the assumption, . This implies that there is a natural surjection which induces an isomorphism . By the assumption that and the nakayama lemma, we see that the kernel of is trivial. This shows that is induced from . The statement that is homology acyclic follows from Proposition 3. ∎
The proof of the following proposition is motivated by a very similar theorem of Ramos for -modules [Ram, Theorem B].
Let be a module generated in finite degrees. Then is homology acyclic if and only if is semi-induced. More generally, if then the graded pieces (successive quotients ) of the natural filtration
are induced (more precisely, is induced from ).
By Proposition 3, if is semi-induced then it satisfies for , and is thus acyclic. The reverse inclusion follows from the second assertion which we now prove by induction on . Note that is concentrated in degree , and injects into (Proposition 2.3). Thus applying to the exact sequence
shows that . By Lemma 3.1, is induced from , and hence acyclic. Thus . The rest follows by induction. ∎
Suppose is semi-induced module generated in degree . Then the graded pieces (successive quotients ) of the natural filtration
are induced (more precisely, is induced from ).
4. The shift theorem
The aim of this section is to prove our main result – the shift theorem.
4.1. The shift and the difference functors I
The category of -vector spaces (and in particular, ) has a symmetric monoidal structure given by the direct sum of vector spaces. It allows us to define a shift functor on -vector spaces (or on ) by
Moreover, for any -linear map , we have a natural transformation given by .
We say that a morphism is of -rank if the dimension of is (clearly, -rank of is at most ). In other words, is the least integer such that there are -morphisms and satisfying . We call any decomposition of the form as above, an -decomposition of . The following lemma is immediate from basic linear algebra.
Let are two -decompositions of . Then there is a unique such that and .
Let be the free -module on morphisms of -rank . Then is a -module in both of the arguments and , and has a natural action of on the right.
We have the following:
is a free -module.
Given a -morphism , the natural map
given by is a split injection of -modules in the variable .
The first two parts are immediate from the previous lemma. Since is an injection, it admits an -linear section (which may not be an injection). This defines a map given by
This map is clearly functorial in and is a section to , finishing the proof. ∎
The functor induces an exact functor , which we again call the shift functor, on given by . An element acts on where the action is induced by . Similarly, there is an action of on .
We have the following:
where is any -module. In particular, shift of an induced module is induced, and shift of a projective -module is projective.
Since every -morphism is of -rank at most , we have an isomorphism . This isomorphism is clearly functorial in . The rest follows from the previous lemma. ∎
Shift of an induced (semi-induced) -module is induced (respectively semi-induced). Category of modules generated (presented) in finite degrees is stable under shift. In particular, .
Exactness of the shift functor and the previous proposition yields the first assertion. The second assertion follows from Proposition 3 and the previous proposition. ∎
Suppose , and be the corresponding natural transformation. If is a -module, then naturally induces a map which is functorial in . We denote the cokernel of this map by . When , we simply denote this cokernel by , or simply if we also have .
Let be a -module. Then is split injective and is an induced module.
If is of -rank then is clearly of -rank . Thus takes the th direct summand of to the th direct summand of , and is functorial in . Thus it suffices to show that the map is split and the cokernel is induced. That it is split is proven in Lemma 4.1(c), and that the cokernel is induced follows from Lemma 4.1(b) and Proposition 3. This proves the result when . The general result follows by observing that tensoring preserves split injections. ∎
The following basic result is easy to establish.
Let and be a -module. Then
The shift commutes with . In particular, .
The kernel of is a torsion module of degree . In particular, is injective if .
4.2. The shift and the difference functors II
We define another shift-like functor which has better formal properties than . We first set some notation. Let be a flag on a vector space given by
We call the stabilizer of in the parabolic subgroup corresponding to and denote it by . The unipotent radical of is the kernel of the natural map
and is denoted by . Fix a maximal flag
In particular, is equal to the dimension of . Set and for . Denote the unipotent radical corresponding to the flag
by . Then given by is clearly a -group, that is, is a functor from to groups. This is in contrast with , which does not define a -group. We define on (or ) by , that is,
It is not hard to see that if is a -module then is a -module. In fact, all we need to check is that for every -morphism and there exists a such that . But one can simply take to be (the last expression makes sense because is a -group) and check that the equation holds. Thus is a functor. Here we have suppressed the choice of flag on . We drop the superscript from (or ) when is of dimension 1.
Suppose we are given an and maximal flags of and such that takes the flag on to an initial segment of the flag on . Any